Definition of groups - subgroups, homomorphism and isomorphism, group representations, irreducible representations, unitary representations

1. Definition of Groups

In mathematics, a group is a fundamental algebraic structure. It consists of a set of elements, along with an operation that combines any two elements to form a third element. This operation must satisfy specific properties. The concept of a group is central to many areas of mathematics, including abstract algebra, number theory, and geometry. It is also widely used in physics, particularly in quantum mechanics, particle physics, and solid-state physics, to describe symmetries.

Formally, a group (G, *) is a set G equipped with a binary operation * that satisfies the following four axioms:

• Closure: For any two elements a and b in G, the result of the operation, a * b, is also an element of G.
• Associativity: For any elements a, b, and c in G, the equation (a * b) * c = a * (b * c) holds true.
• Identity element: There exists an element e in G such that for every element a in G, the equation e * a = a * e = a holds. This element e is called the identity element.
• Inverse element: For each element a in G, there exists an element a^-1 in G such that a * a^-1 = a^-1 * a = e, where e is the identity element. The element a^-1 is called the inverse of a.

If the operation * is also commutative (i.e., a * b = b * a for all a, b in G), then the group is called an abelian group or a commutative group.

Example: The set of integers Z with the operation of addition (+) forms an abelian group.

• Closure: The sum of two integers is always an integer.
• Associativity: (a + b) + c = a + (b + c) for all integers a, b, c.
• Identity element: The integer 0 is the identity element, as 0 + a = a + 0 = a for any integer a.
• Inverse element: For any integer a, its inverse is -a, since a + (-a) = (-a) + a = 0.

Example in Physics: Consider the set of rotations in a 2D plane about the origin. The operation is composition of rotations. This set forms a group under composition.

2. Subgroups

A subgroup is a subset of a group that is itself a group under the same operation as the parent group. To show that a non-empty subset H of a group G is a subgroup, we need to verify that H satisfies the group axioms. However, a simpler criterion, known as the subgroup test, can be used.

Subgroup Test: Let G be a group and H be a non-empty subset of G. H is a subgroup of G if and only if for any two elements a and b in H, the element a * b^-1 is also in H.

Alternatively, if G is a finite group, H is a subgroup if it is non-empty and for all a, b in H, a * b is in H. This is because in a finite group, the existence of inverses is implied by closure and associativity if the set is finite.

Example: Consider the group of integers under addition (Z, +). The set of even integers, E = {..., -4, -2, 0, 2, 4, ...}, is a subgroup of (Z, +).

• Let a = 2m and b = 2n be two even integers in E.
• Then a * b^-1 = 2m + (-2n) = 2(m - n).
• Since (m - n) is an integer, 2(m - n) is an even integer, which is in E.
• Thus, E is a subgroup of (Z, +).

Example in Physics: In molecular symmetry, the point group of a molecule describes its symmetry operations. The set of all symmetry operations forms a group. Subgroups of this point group correspond to subsets of symmetry operations that also form a group, often representing simpler symmetries or symmetries of fragments of the molecule.

3. Homomorphism and Isomorphism

Homomorphism and isomorphism are concepts that describe relationships between different groups. They allow us to compare the structure of groups.

3.1 Homomorphism

A homomorphism is a mapping between two groups that preserves their structure. Specifically, a function f: G → G' is a group homomorphism if for all elements a and b in G, the following condition holds: f(a * b) = f(a) *' f(b) where * is the operation in G and *' is the operation in G'.

In simpler terms, applying the operation in the first group and then mapping the result is the same as mapping the elements first and then applying the operation in the second group.

Properties of Homomorphisms:

• A homomorphism maps the identity element of G to the identity element of G'. (f(e_G) = e_G')
• A homomorphism maps the inverse of an element in G to the inverse of the mapped element in G'. (f(a^-1) = (f(a))^-1)

Example: Consider the homomorphism from the group of integers under addition (Z, +) to the group of even integers under addition (E, +), defined by f(n) = 2n.

• Let a, b be integers.
• f(a + b) = 2(a + b) = 2a + 2b.
• f(a) + f(b) = 2a + 2b.
• Since f(a + b) = f(a) + f(b), this is a homomorphism.

3.2 Isomorphism

An isomorphism is a special type of homomorphism that is also a bijection (both one-to-one and onto). If there exists an isomorphism between two groups G and G', then the groups are said to be isomorphic.

A function f: G → G' is an isomorphism if:

• f is a homomorphism.
• f is one-to-one (injective): If f(a) = f(b), then a = b.
• f is onto (surjective): For every element y in G', there is at least one element x in G such that f(x) = y.

Isomorphic groups are essentially the same from an algebraic perspective. They have the same structure, even if their elements are different. We say that two groups are structurally identical if they are isomorphic.

Example: The group of integers modulo 4 under addition, (Z₄, +), and the Klein four-group V = {e, a, b, ab} under a specific multiplication rule are isomorphic.

Example in Physics: Isomorphism is crucial for understanding symmetries. If two different physical systems exhibit the same symmetry group, their underlying dynamical behavior related to these symmetries will be mathematically identical. For instance, different physical laws might be described by the same Lie group.

4. Group Representations

In physics, especially in quantum mechanics and particle physics, groups themselves are often abstract mathematical entities. To study their physical relevance, we represent these abstract groups as groups of matrices. A group representation is a way to map the elements of an abstract group to invertible linear transformations (matrices) on a vector space, in such a way that the group structure is preserved.

Formally, a representation of a group G on a vector space V is a homomorphism D: G → GL(V), where GL(V) is the group of all invertible linear transformations on V. If V is a finite-dimensional vector space, say of dimension n, then GL(V) is isomorphic to the group of n×n invertible matrices, denoted by GL(n, C) (for complex vector spaces) or GL(n, R) (for real vector spaces).

So, a representation of G is a mapping D from each element g in G to an n×n matrix D(g) such that: D(g₁ * g₂) = D(g₁) D(g₂) for all g₁, g₂ in G. Here, * denotes the group operation in G, and the product on the right is matrix multiplication.

The vector space V is called the representation space. The dimension of the representation space is the dimension of the matrices.

Example: Consider the cyclic group C₃ = {e, a, a²} where a³ = e. We can represent this group using 1x1 matrices (complex numbers). Let D(e) = [1]. Let D(a) = [ω], where ω = e^2πi/3 is a complex cube root of unity. Then D(a²) = [ω²]. Check: D(a * a) = D(a²) = [ω²]. D(a)D(a) = [ω][ω] = [ω²]. This holds. D(a * a²) = D(a³) = D(e) = [1]. D(a)D(a²) = [ω][ω²] = [ω³] = [1]. This also holds. This is a one-dimensional representation.

Example in Physics: In quantum mechanics, the states of a system can be represented by vectors in a Hilbert space. The symmetry operations of the system (e.g., rotations, translations) act as operators on this Hilbert space. These operators form a representation of the symmetry group. For example, the angular momentum operators in quantum mechanics are related to the representation of the rotation group SO(3).

5. Irreducible Representations (Irreps)

A representation D of a group G on a vector space V is called reducible if there exists a non-trivial subspace W of V (i.e., W ≠ {0} and W ≠ V) such that W is invariant under the action of all the operators D(g) for g in G. This means that if v is in W, then D(g)v is also in W for all g in G.

If such a non-trivial invariant subspace exists, the representation can be decomposed into smaller representations acting on these invariant subspaces.

A representation is called irreducible (or an irrep) if its only invariant subspaces are the trivial ones: the zero subspace {0} and the entire vector space V. Irreducible representations are the fundamental building blocks of all representations. Any representation can be decomposed into a direct sum of irreducible representations.

Decomposition of Reducible Representations: If a representation D is reducible, it can be written as a direct sum of irreducible representations: D = D₁ ⊕ D₂ ⊕ ... ⊕ D_k where D_i are irreducible representations.

Example: Consider the group of rotations in 3D space, SO(3). The vector representation of SO(3) (where vectors transform as expected under rotation) is irreducible. However, the representation of SO(3) on the space of spherical harmonics Y_lm for a fixed angular momentum quantum number l is irreducible, but for a space containing spherical harmonics for all l, it becomes reducible.

Schur's Lemma: A fundamental theorem related to irreducible representations states that if D₁ and D₂ are irreducible representations of a group G on vector spaces V₁ and V₂ respectively, and if there is a linear map T: V₁ → V₂ such that T D₁(g) = D₂(g) T for all g in G, then either T is the zero map, or T is an isomorphism. If V₁ = V₂ and D₁ = D₂, then T must be a scalar multiple of the identity (T = cI). This lemma is extremely important in proving properties of irreps.

Character of a Representation: The character of a representation D at an element g is defined as the trace of the matrix D(g): χ(g) = Tr(D(g)). Characters are useful because:

• The character is invariant under change of basis (equivalent representations have the same character).
• The character of a representation is the sum of the characters of its irreducible components.
• The characters of irreducible representations form an orthogonal basis for the space of class functions on the group.

Example in Physics: In particle physics, elementary particles are classified according to the irreducible representations of symmetry groups like SU(2) (related to spin) and SU(3) (related to the strong nuclear force, flavor, and color). For example, quarks and gluons are classified under specific irreps of SU(3).

6. Unitary Representations

A unitary representation is a representation where the matrices are unitary. A representation D: G → GL(V) is unitary if the vector space V is a Hilbert space (i.e., it has an inner product), and for every g in G, the linear transformation D(g) is a unitary operator.

An operator U on a Hilbert space V is unitary if its adjoint U^† satisfies U^†U = UU^† = I, where I is the identity operator. In terms of matrices, this means D(g)^†D(g) = D(g)D(g)^† = I for all g in G.

Unitary representations are particularly important in physics because:

• They preserve the probabilistic interpretation of quantum mechanics. The inner product of quantum states (which determines probabilities) must be preserved under symmetry transformations.
• They ensure that probabilities are conserved over time.

Theorem: Every finite-dimensional representation of a finite group is equivalent to a unitary representation. This means that even if a given representation is not explicitly unitary, we can find a change of basis that makes it unitary.

Example: Consider the rotation group SO(3). The representation describing the transformation of spatial vectors is unitary. If we consider the representation of angular momentum operators L_x, L_y, L_z in quantum mechanics, they satisfy [L_i, L_j] = iħ ε_ijk L_k. The operators exp(-iθL_n/ħ) represent rotations and are unitary operators, forming a unitary representation of the rotation group.

Example in Physics: In quantum field theory, fields are often associated with representations of the Poincaré group (which describes spacetime symmetries). For a consistent theory, these representations must be unitary, ensuring that probabilities are conserved and that the theory is physically meaningful.